Instrumental variables

class: center, middle, inverse, title-slide

.title[
# <a href="https://github.com/mca91/kausalanalyse_slides/raw/main/IV/IV.pdf">Instrumental variables</a>
]
.author[
### Christoph Hanck
]
.date[
### Summer 2023
]

---

# Isolating variation

.vcenter[
.blockquote[
### Definition: Instrumental variable
Instead of trying to strip away all the undesirable variation using controls, an instrumental variable (IV) is a source of variation that allows to isolate just the interested front-door path. IV designs attempt to mimic a randomized experiment, but using statistics and opportune settings instead of actually being able to influence or randomize anything.

For instrumental variables to work, two assumptions must be satisfied:

- Relevance of the instrument
- Validity of the instrument

]]

---
# Isolating variation
 
**A typical setting with randomization**

<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/IV/renderthis_3dce54b5a3e_files/figure-html/unnamed-chunk-2-1.png" alt="1: Causal Diagram" width="65%" style="display:block; margin-right:auto; margin-left:auto; margin-top:50px;" />
1: Causal Diagram
</div>

---
# Isolating variation

An IV design works in as above:

- An IV design does not remove the requirement to closing back doors in order to identify an effect

- IV shifts the requirement to something simpler: Instead of needing to close the back doors between `$Treatment$` and `$Outcome$` it becomes enough to 
    
    ...close the back doors between `$Randomization$` and `$Outcome$`.
    
    ...close the front doors between `$Randomization$` and `$Outcome$` that do not go through `$Treatment$`.

(If there are any, respectively.)

---
# Isolating variation

**Steps**

1.  Use the IV to explain the treatment
2.  Remove any part of the treatment that is not explained by the instrument
3.  Use the instrument to explain the outcome
4.  Remove any part of the outcome that is not explained by the instrument
5.  Look at the relationship between the remaining, instrument-explained part of the outcome and the remaining, instrument-explained part of the treatment

---
# Isolating variation
 
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/IV/renderthis_3dce54b5a3e_files/figure-html/unnamed-chunk-3-1.png" alt="2: Instrumental variables with a binary instrument" width="100%" height="100%" style="display:block; margin-right:auto; margin-left:auto;" />
2: Instrumental variables with a binary instrument
</div>

---
# Assumptions for IV
 
**Relevance of the instrument**
 
- The key idea of IV is to use that part of treatment `$X$` that is explained by the instrument `$Z$` is used to explain the outcome `$Y$`. In its basic form, IV gives us `$\frac{\text{Cov}(Z,\,Y)}{\text{Cov}(Z,\,X)}$`.

- If no part of `$X$` is explained by `$Z$` then `$\text{Cov}(Z,\,X)=0$` and hence we get `$\frac{\text{Cov}(Z,\,Y)}{0}$`

&rarr; IV does not work!

**Validity of the instrument**

Validity is the assumption that the instrument `$Z$` is has no open back doors of its own:

Any paths between the instrument `$Z$` and the outcome `$Y$` must either pass through the treatment `$X$` or be closed.

---
# Assumptions for IV
 
<div class="figure" style="text-align: center">
<img src="data:image/png;base64,#/Users/martin/git_projects/KA_slides/IV/renderthis_3dce54b5a3e_files/figure-html/unnamed-chunk-4-1.png" alt="3: A causal diagram amenable to IV estimation" width="65%" style="display:block; margin-right:auto; margin-left:auto;" />
3: A causal diagram amenable to IV estimation
</div>
---
#Assumptions for IV
 
**Validity of the instrument**
 
There are a few paths from `$Z$` to `$Y$`
- `$Z \rightarrow X \rightarrow Y$`
- `$Z \rightarrow X \leftarrow D \rightarrow Y$`
- `$Z \leftarrow C \rightarrow X \rightarrow Y$`
- `$Z \leftarrow C \rightarrow X \leftarrow D \rightarrow Y$`
- `$Z \leftarrow B \rightarrow Y$`
- `$Z \rightarrow A \rightarrow Y$`

---
# Assumptions for IV
 
**Validity of the instrument**
 
- Assume that we cannot control for the endogeneity through `$D$` and we want to use `$Z$` as an IV
 
 &rarr; We require all open paths from `$Z$` to `$Y$` to contain `$X$`.
 
- Putting paths that contain `$X$` to the side, that gives
 - `$Z \leftarrow B \rightarrow Y$`
 - `$Z \rightarrow A \rightarrow Y$`

&rarr; These two paths are problematic (why?) and must be shut down for `$Z$` to be valid!

- So for the validity assumption to hold for `$Z$`, we need to control for `$A$` and `$B$`.
 
---
#Assumptions for IV
 
**How realistic is validity?**

.blockquote[
###Example: Effect of population’s health on economic growth in that country

- Acemoglu and Johnson (2007) use the timing of when new medical technology (like vaccines for certain diseases) is introduced as an instrument for a country’s health (specifically, mortality from certain diseases)

- They find that changes in health in a given year driven by changes in medical technology that year have a negative effect on a country’s economic growth
]

---
# Assumptions for IV
 
**How realistic is validity?**

.blockquote[
### Example: Effect of population’s health on economic growth in that country
- Bloom et al. (2014) state that the original study did not account for the possibility that changes in health may have long-term effects on growth:

The healthier the population, the less a new medical technology can further improve mortality rates.

- Effectiveness of medical technology is related to pre-existing health, which is related to pre-existing economic factors, which is related to current economic growth.

&rarr; Back door!

Fixing the validity issue by controlling for pre-existing factors, Bloom et al. find a *positive* effect!

]

---
# Canonical designs
 
The use of IV means requires *clever design*: finding causal inference using good instruments in situations with numerous back doors.

Clever instruments are often highly context-dependent. One good example of a canonical IV design is **judge assignment**.

.blockquote[
###Example: Judge assignment (Aizer and Doyle Jr. 2015)

- In many court systems, the process of assigning a judge to your trial is more or less random. This is important because some judges are harsher than others.

- This means `$JudgeHarshness$` can act as an instrument for `$Punishment$`

]

---
# Canonical designs
.vcenter[
.blockquote[
###Example: Judge Assignment

- By estimating the harshness of each judge using their prior rulings, the harshness of the judge becomes an instrument for the punishment.

- This can be used anywhere where there is the system of randomly-assigned judges and the interest is to know the impact of harsher punishments (or even being judged guilty) on some later outcome.
]]

---
# IV estimators
 
**IV estimators can be computed using OLS**

Instrumental Variable can be estimated using Two-stage least squares (2SLS). The method uses two regressions to estimate an IV regresssion model
 
 - The **first stage** uses the instrument `$Z$` (and controls `$W$`) to predict the treatment variable `$X$`, `$$X = \gamma_0 + \gamma_1Z + \gamma_2W + \nu$$` 
 
 - In the **second stage**, the predicted (explained) values of the treatment variable `$\widehat{X}$` from the first stage are used to predict the outcome `$Y$` in the second stage (along with the controls), `$$Y = \beta_0 + \beta_1\widehat{X} +\beta_2W + \epsilon$$`
 
---
# IV estimators
 
**Interpretation of 2SLS**
 
- 2SLS produces a *ratio of effects*, dividing the effect of `$Z$` on `$Y$` by the effect of `$Z$` on `$X$`

- It asks for each movement in `$X$` we get by changing `$Z$`, how much movement in `$Y$` does that lead to?'

&rarr; Since `$Z$` has no back doors, this should give us the causal effect of `$X$` on `$Y$`.

**Advantages and disadvantages of 2SLS**

- Easy to estimate and flexible: Adding more instruments to the first stage is a no-brainer (adding more treatment variables is less easy)
- Bad small-sample performance. For real data, the relationship between `$Z$` and the 'cleansed' part of `$Y$` is going to be somewhat nonzero, just by random chance 
- 2SLS also does not perform particularly well when the errors are heteroskedastic

---
# IV estimators

**Generalized Method of Moments (GMM)**

Using assumptions and theory, construct statistical moments (means, variances, covariances, etc.) that should have certain values. Use sample versions of these moments to estimate the quantity of interest

.blockquote[
### Example: The sample mean is a GMM estimator of the expected value

- To use GMM to estimate the expected value of a variable `$Y$`, it can be said that the difference between the estimator `$\mu$` and the population expected value `$E(Y)$` should be zero

&rarr; `$\mu - E(Y) = 0$` is a moment condition for the estimation.

- Replace `$E(Y)$` with its sample value `$\frac{1}{N}\sum Y$` and solve the equation to get `$\widehat{\mu} = \frac{1}{N}\sum Y$`

- GMM will pick the `$\mu$` that makes the moment condition true, on average.
]

---
# IV estimators
 
**Overidentification**

- When the number of instruments is bigger than the number of treatment/endogenous variables, we say that the model is **overidentified**

**When the model is overidentified, 2SLS and GMM will diverge!**

- GMM will have less sampling variation (and thus smaller standard errors) under heteroskedasticity than 2SLS, even if heteroskedasticity-robust standard errors are used

---
# IV estimation and treatment effects
 
For a standard estimator like 2SLS or GMM with one treatment/endogneous variable and *one* instrument, the weights are what the individual effect of the instrument would be in the first stage.
 
.blockquote[
### Example: Effect of exercise on blood pressure
A recent set of TV ads encourages people to exercise more. Exposure to the advertisements is used as an instrument for how much a person exercises.

|Name|Effect of Ads on Exercise Hours| Effect of Exercise Hours on Blood Pressure|
|:----|:-----:|:-----:|
|Jakeila| `$0.5$` | `$-2$` |
|Kyle| `$0.25$` | `$-8$` |
|Li| `$0.00$` | `$-10$` |

]